1.3 segs & their meas. ppt 1
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The document outlines several postulates about segments and their measures on a number line, including that the distance between two points can be found by taking the absolute value of the difference between their coordinates, and that if a point B is between points A and C, then the sum of the lengths of segments AB and BC will equal the length of segment AC. Examples are provided to demonstrate applying these postulates, such as using the distance formula or Pythagorean theorem to find lengths of segments. Key terms like congruent segments are also defined.
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1.3 segs __their_meas.
The document outlines several postulates about segments and their measures on a number line, including that the distance between two points can be found by taking the absolute value of the difference between their coordinates, and that if a point B is between points A and C, then the sum of the lengths of segments AB and BC will equal the length of segment AC. Examples are provided to demonstrate using the segment addition postulate and finding lengths of segments using given information. Key terms like congruent segments and the Pythagorean theorem are also defined.
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The document provides examples and steps for writing equations of perpendicular bisectors in the coordinate plane, including finding the midpoint and slope of a segment to determine the equation of the perpendicular bisector in point-slope form, with an example working through writing the equation of the perpendicular bisector of the segment from (6, -5) to (10, 1).
Section1.3a
This document defines several key geometry terms:
- A segment is defined as two points on a line connected by all points between them. The endpoints are the defining points of the segment.
- A ray is one endpoint point on a line extending in one direction. Opposite rays extend from the same endpoint in opposite directions.
- The length of a segment is the distance between its two endpoints. On a number line, length is the absolute value of the difference between the coordinates.
- Other terms defined include postulates, congruent, congruent segments, midpoint of a segment, and segment bisector. Practice problems are provided to reinforce the new definitions.
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The document defines key terms used in geometry proofs including: adjacent angles, conditionals, converse, counterexample, and equivalence relations. It also outlines properties of equality and congruence such as: reflexive, symmetric, transitive, addition, subtraction, multiplication, and division properties. Finally, it discusses two-column proofs and Euler diagrams as methods for logical reasoning about geometric statements.
Bfpt
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2) The document provides two proofs of Sperner's Lemma: one using a "paths through rooms" analogy and another using edge labelings.
3) Brouwer's Fixed Point Theorem states that any continuous function from a triangle to itself must have a point that is fixed. The document proves this for n=2 dimensions using Sperner's Lemma and barycentric coordinates.
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This document provides examples and exercises demonstrating the use of angle bisectors in triangles to solve problems. Example 1 shows using the angle bisector theorem to find the measure of an angle. Example 2 shows a soccer goalie positioned on the angle bisector and being equidistant from both goalposts. Example 3 uses the converse of the angle bisector theorem to find the value of x that positions a point on the angle bisector. Example 4 uses angle bisector and triangle congruency theorems to find the length of a side of an inscribed triangle. The guided practice and exit slip exercises provide additional problems for students to practice finding values using angle bisectors.
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The document outlines several postulates about segments and their measures on a number line, including that the distance between two points can be found by taking the absolute value of the difference between their coordinates, and that if a point B is between points A and C, then the sum of the lengths of segments AB and BC will equal the length of segment AC. Examples are provided to demonstrate using the segment addition postulate and finding lengths of segments using given information. Key terms like congruent segments and the Pythagorean theorem are also defined.
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The document provides examples and steps for writing equations of perpendicular bisectors in the coordinate plane, including finding the midpoint and slope of a segment to determine the equation of the perpendicular bisector in point-slope form, with an example working through writing the equation of the perpendicular bisector of the segment from (6, -5) to (10, 1).
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This document defines several key geometry terms:
- A segment is defined as two points on a line connected by all points between them. The endpoints are the defining points of the segment.
- A ray is one endpoint point on a line extending in one direction. Opposite rays extend from the same endpoint in opposite directions.
- The length of a segment is the distance between its two endpoints. On a number line, length is the absolute value of the difference between the coordinates.
- Other terms defined include postulates, congruent, congruent segments, midpoint of a segment, and segment bisector. Practice problems are provided to reinforce the new definitions.
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The document defines key terms used in geometry proofs including: adjacent angles, conditionals, converse, counterexample, and equivalence relations. It also outlines properties of equality and congruence such as: reflexive, symmetric, transitive, addition, subtraction, multiplication, and division properties. Finally, it discusses two-column proofs and Euler diagrams as methods for logical reasoning about geometric statements.
Bfpt
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2) The document provides two proofs of Sperner's Lemma: one using a "paths through rooms" analogy and another using edge labelings.
3) Brouwer's Fixed Point Theorem states that any continuous function from a triangle to itself must have a point that is fixed. The document proves this for n=2 dimensions using Sperner's Lemma and barycentric coordinates.
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This document provides examples and exercises demonstrating the use of angle bisectors in triangles to solve problems. Example 1 shows using the angle bisector theorem to find the measure of an angle. Example 2 shows a soccer goalie positioned on the angle bisector and being equidistant from both goalposts. Example 3 uses the converse of the angle bisector theorem to find the value of x that positions a point on the angle bisector. Example 4 uses angle bisector and triangle congruency theorems to find the length of a side of an inscribed triangle. The guided practice and exit slip exercises provide additional problems for students to practice finding values using angle bisectors.
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2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
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This document discusses coordinates in space and three-dimensional coordinate geometry. It introduces points, lines, and planes in three-dimensional space and how they are represented using ordered triples of real numbers called coordinates. It describes how the three mutually perpendicular coordinate axes divide space into eight octants. It provides formulas for finding distances between points, section formulas, midpoints, and centroids. It also discusses direction cosines and direction ratios as ways to represent the direction of lines in space, and provides formulas for finding angles between lines based on their direction cosines or direction ratios.
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1.3 segs & their meas. ppt 1
- 1. 1.3 Segments & their Measures p. 17
- 2. Postulates • Rules that are accepted as true with having to be proven. • Sometimes they are called axioms.
- 3. Postulate 1: Ruler post. • The points on a line can be matched with those on the real # line. • The real # that corresponds to a point is the coordinate of the point. • If you find the difference between the coordinates of 2 points, then take the absolute value, you will have the distance between the points.
- 4. Ruler post. (continued) A B x1 x2 • The length of AB can be found by x2-x1 • **The symbol for the length of AB is AB.
- 5. Example: Find AB. A B Point A is at 1.5 and B is at 5. So, AB = 5 - 1.5 = 3.5
- 6. Is Alex between Ty and Josh? Yes! Ty Alex Josh No, but How about now? why not? In order for a point to be between 2 others, all 3 points MUST BE collinear!!
- 7. Post. 2: Segment Addition Post. • If B is between A & C, then AB + BC = AC. • If AB + BC = AC, then B is between A & C. C B A
- 8. Ex: if DE=2, EF=5, and DE=FG, find FG, DF, DG, & EG. D E F G FG=2 DF=7 DG=9 EG=7
- 9. Reminders: • Pythagorean Theorem – a2+b2=c2 a & b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. • Distance formula – ( x2 − x1 ) 2 + ( y2 − y1 ) 2 (x1,y1) & (x2,y2) are the 2 points.
- 10. Congruent ( ≅ ) Segments • Segments that have the same length. If AB & XY have the same length, Then AB=XY, but AB ≅ XY Symbol for congruent
- 11. Assignment